diff --git a/Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf b/Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf new file mode 100755 index 0000000..f99fa41 Binary files /dev/null and b/Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf differ diff --git a/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg b/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg new file mode 100755 index 0000000..6cdd5c4 --- /dev/null +++ b/Lecture-7-slides-for-printing-slide-13-gap-parameter.svg @@ -0,0 +1,148 @@ + +image/svg+xmlTTc= 1.76kBTc diff --git a/references.bib b/references.bib index ea7ac07..a32fa26 100755 --- a/references.bib +++ b/references.bib @@ -111,7 +111,7 @@ Results from previous neutron measurements are found to be consistent with the X @book{annett, edition = {1}, - series = {{OXFORD} {MASTER} {SERIES} {IN} {CONDENSED} {MATTER} {PHYSICS}}, + series = {Oxford Master Series in Condensed Matter Physics}, title = {Superconductivity, {Superfluids}, and {Condensates}}, volume = {5}, isbn = {978-0-19-850756-7 0-19-850756-9}, diff --git a/superconductivity_assignment5_kvkempen.tex b/superconductivity_assignment5_kvkempen.tex index f0cfabb..a1d0979 100755 --- a/superconductivity_assignment5_kvkempen.tex +++ b/superconductivity_assignment5_kvkempen.tex @@ -58,12 +58,71 @@ with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of state A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course. The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is \[ - -E = 2\hbar\omega_De^{-1/\lambda}. + -E = 2\hbar\omega_De^{-1/\lambda} =: \Delta_0, \] +which is also called the gap parameter $\Delta_0$ at zero temperature for BCS. In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$. +It should be noted that this weak limit also means that the gap is smaller than the thermal energy of the highest excited energy phonon, which corresponds to the Debye temperature +\[ + \Delta < k_B\Theta_D. +\] It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$. -We will look at a way to express the critical temperature in terms we can derive, and than look at the values that maximize this critical temperature whilst still following BCS theory. +We will look at a way to express the critical temperature in terms we can derive, and then look at the values that maximize this critical temperature whilst still following BCS theory. +From the derivation of the BCS coherent state, this gap parameter at finite temperature is found. +There is a temperature dependence $\Delta(T)$ as in figure \ref{fig:gap-T}. +\begin{figure} + \centering + \includegraphics[width=.4\textwidth]{Lecture-7-slides-for-printing-slide-13-gap-parameter.pdf} + \caption{By taking the gap parameter to zero, we find the critical temperature. Figure from the slides of lecture 7.} + \label{fig:gap-T} +\end{figure} +For larger temperatures, thermal energy is increased, and less energy is required to break up Cooper pairs, thus degrading the superconductivity. +This puts a limit $T_c$. + +We will mostly follow the derivation by Waldram \cite[paragraph 7.9, mostly p.128--130]{waldram}. +The superconducting state breaks down at high temperature, at which also $\Delta$ vanishes so that the gap parameter is a good order parameter for the state. + +Let's consider the gap parameter +\[ + \Delta_{\vec{k}} = -\sum_{\vec{k'}}(1-2f_{\vec{k'}})u_{\vec{k'}}v_{\vec{k}}V_{\vec{k'}\vec{k}}, +\] +with $u$ and $v$ occupation functions for the BCS state, $f$ the Fermi occupation number, and $V$ the potential between the states. +Minimizing $\Delta_{\vec{k}}$ and taking that $V_{\vec{k'}\vec{k}} = -V$ is constant gives us a self-consistent relation for the gap parameter. +We also recognize that the states that we sum over all all those states such that they have smaller energy than the highest excited phonon. +\[ + \Delta_{\vec{k}} = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{\Delta_{\vec{k'}}}{2E_{\vec{k'}}}. +\] +Now the right-hand side is independent of $\vec{k}$ but does contain $\Delta_{\vec{k'}}$. +We can thus conclude that the gap parameter should be constant over all states $\vec{k}$! +That means we can divide both sides by it, giving us +\[ + 1 = V\sum_{\epsilon_{\vec{k'}}}(1-2f_{\vec{k'}})\frac{1}{2E_{\vec{k'}}}. +\] +Converting the equation to an integral, and substituting in $f(E) = [\exp{(E/(k_BT))}+1]^{-1}$ and $E = \sqrt{\epsilon^2 + \Delta(T)^2}$ yields +\[ + 1 = 2g(\epsilon_F)V\int_0^{k_B\Theta_D}\frac{1-2[\exp{(E/(k_BT))}+1]^{-1}}{2\sqrt{\epsilon^2 + \Delta(T)^2}} \textup{d}\epsilon. +\] + +I believe Waldram that one could find that +\[ + T_c = 1.14\Theta_D\exp{(-1/(g(\epsilon_F)V))} = 1.14\Theta_D\exp{(-1/(\lambda)} +\] +from this nice equation. + +As limiting value, we take $\lambda = 0.3$, as was posed as a reasonable limit for the weak coupling by Alix in lecture 7, +although Waldram \cite{waldram} thinks it is more like $\lambda \approx 0.4$. + +For metals, Waldram thinks $\Theta_D \leq \SI{300}{\kelvin}$ is a good limit. +This leads to our final maximum +\[ + T_c \leq 1.14 \cdot 300 \cdot \exp{(-1/0.4)} \approx \SI{28}{\kelvin}. + %https://www.wolframalpha.com/input?i=1.14*300*e%5E%28-1%2F.4%29 +\] +(Using $\lambda = 0.3$ yields an even lower $T_c \leq \SI{12}{\kelvin}$.) + +For larger $T_c$ values, larger binding of Cooper pairs would be needed to overcome the thermal energy. +This means our assumption of weak coupling breaks down, making most of the derivation invalid without further arguments. \section{Energy gap $\Delta$ et al.}